The study of the virial relation in the context of general relativity has a long history. Beginning with the early work of Lindquist [1]} on the covariant collision-less Boltzmann equation, Jackson [2]} published a first formulation of general relativistic virial theorem using the kinetic theory approach for a case of static spherically symmetric structure. His approach has been extended to the virial theorem in the modified gravity theories[3]}. There has been another approach to relativistic virial theorem using some of the Einstein equations or its foliation to define energy and derive an equation similar to the classical virial relation. Authors in [4]} derive a relation including terms to be interpreted in the Newtonian limit as potential and kinetic energy using the ADM foliation for a static asymptotically flat space-time, based on the older paper [5]}. Recently Meyer et al. [6]} have written a relativistic version of the virial theorem for the \(\Lambda CDM\)  cosmologies, using a heuristic approach in defining a potential and kinetic energy term.
There are different aspects of the context of general relativity, or the formulation of the virial theorem in general relativity. Usually, in speaking of a relativistic condition, one think of the strong gravity regime. There is however another area in the cosmology and astrophysics where general relativistic effects may come in without being in a strong gravity regime. This happens when one is dealing with non-local or large scale phenomena [7]}, [8]}, [9]}, [10]}, [11]}. The expansion of
the universe, as an example, is a general relativistic effect at large scale in a very weak gravity regime: assuming any non-zero density, however small it may be, Einstein equations lead to a non-static expanding universe, in contrast to the Newtonian gravity. The general relativistic virial theorem in the cosmological context, therefore, may reveal novel features not expected from the semi-relativistic static cases. The main motivation of our current study is to find out if one has to expect novel general relativistic features while interested in large scale problems within astrophysics and cosmology.
